Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers

For totally positive algebraic integers of degree , we consider the set of values of , where is the Mahler measure of . C. J. Smyth has found the four smallest values of and conjectured that the fifth point is . We prove that this is so and, moreover, we give the sixth point of .

     

Representation of integers by positive-definite quadratic forms in totally real algebraic number fields

The results of Gundlach (K. B. Gundlach, Acta Math.,92, 309–345 (1954)) are extended to forms with an odd number of variables. We obtain for alln4 asymptotic formulae for the number of integral representations of numbers by totally positiven-ary forms in totally real algebraic number fields. The remainder term is of the form . The necessary concepts and results about Hilbert modular forms of half integral weight are developed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 32–46, 1983.     

Size of algebraic integers

Translated from Matematicheskie Zametki, Vol. 49, No. 4, pp. 152–154, April, 1991.     

Representations of positive integers

The problem of representing integers as sums of terms of certain type is actual in number theory and its applications. We are interested in the average length of these expansions and the required number of auxiliary calculations. The paper deals with DBNS, chains, and the polyadic (factorial) expansions of positive integers.     

Irreducible algebraic integers in short intervals

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x ^θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x ₀. The constant x ₀ effectively depends on K and θ. The author was supported in part by the grant no. N N201 1482 33 from the Polish Ministry of Science and Higher Education.     

Hochschild homology of rings of algebraic integers

The purpose of this paper is to remind that the André-Quillen homology theory is a very useful tool to study Hochschild homology of commutative algebras. We choose as an example a computation of Hochschild homology of Dedekind extensions.     

On the distribution of irreducible algebraic integers

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2.     

On the distribution of certain algebraic integers

Mathematische Zeitschrift -     

On the distribution of irreducible algebraic integers

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2. Author’s address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland The author was supported in part by KBN Grant # N N201 1482 33.     

On the partial quotients of algebraic integers

It is shown that there are algebraic integers, with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Davenport for algebraic numbers.     

Generating sets for rings of algebraic integers

Let R be the ring of integers of some finite algebraic extension of the rationals Q of degree n. A necessary and sufficient condition for s elements of R to be an R-basis is given, in terms of the Hermite normal form of a certain n ×ns integral matrix depending on the elements, and on the structure constants of R.     

Conjugate algebraic integers in real point sets

Mathematische Zeitschrift -     

Totally real algebraic integers in small intervals

We show that there are irreducible monic polynomials having all roots in an interval of length close to 4. These are obtained by perturbing the coefficients of the respective Chebyshev polynomials. In particular, we obtain that our earlier lower bound for the house of totally real algebraic integers is sharp up to a logarithmic factor. Partially supported by the Lithuanian State Science and Studies Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 307–312, July–September, 2000.     

The approximation of a totally positive band matrix by a strictly banded totally positive one

Every nonsingular totally positive m-banded matrix is shown to be the product of m totally positive one-banded matrices and, therefore, the limit of strictly m-banded totally positive matrices. This result is then extended to (bi)infinite m-banded totally positive matrices with linearly independent rows and columns. In the process, such matrices are shown to possess at least one diagonal whose principal sections are all nonzero. As a consequence, such matrices are seen to be approximable by strictly m-banded totally positive ones.